偏微分方程的數(shù)值近似法

內(nèi)容概要

out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is our primary concern. Many kinds of problems are addressed: linear and nonlinear, steady and time-dependent, having either smooth or non-smooth solutions. Besides model equations, we consider a number of (initial-) boundary value problems of interest in several fields of applications. Part I is devoted to the description and analysis of general numerical methods for the discretization of partial differential equations.

書籍目錄

Part I.Basic Concepts and Methods for PDEs‘ Approximation   1. Introduction     1.1 The Conceptual Path Behind the Approximation     1.2 Preliminary Notation and Function Spaces     1.3 Some Results About Sobolev Spaces     1.4 Comparison Results   2. Numerical Solution of Linear Systems     2.1 Direct Methods    2.2 Generalities on Iterative Methods     2.3 Classical Iterative Methods     2.4 Modern Iterative Methods     2.5 Preconditioning    2.6 Conjugate Gradient and Lanczos like Methods for Non-Symmetric Problems    2.7 The Multi-Grid Method    2.8 Complements  3.Finite Element Approximation    3.1 Triangulation    3.2 Piecewise-Polynomial Subspaces    3.3 Degrees of Freedom and Shape Functions    3.4 The Interpolation Operator    3.5 Projection Opeators    3.6 Complements  4.Polynomial Approximation    4.1 Orthogonal Polynomials    4.2 Gaussian Quadrature and Interpolation    4.3 Chebyshev Expansion    4.4 Legendre Expansion    4.5 Two-Dimensional Extensions    4.6 Complements  5.Galerkin,Collocation and Other Methods    5.1 An Abstract Reference Boundary Value Problem    5.2 Galerkin Method    5.3 Petrov-Galerkin Method    ……Part Ⅱ.Approximation of Boundary Value Problems  6.Elliptic Problems:Approximation by Galerkin and Collocation Methods  7.Elliptic Prblems:Approximation by Mixed and Hybrid Methods  8.Steady Advection-Diffusion-Problems  9.The Stokes Problem  10.The Steady Navier-Stokes ProblemPart Ⅲ.Approximation of Initial-Boundary Value Problems  11.Parabolic Problems  12.Unsteady Advection-Diffusion Problems  13.The Unsteady Navier-Stokes Problem  14.Hyperbolic ProblemsReferencesSubject Index

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